P colonies working in the maximally parallel and in the sequential mode

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We consider P colonies as introduced in Kelemen et al. (2005) and investigate their computational power when working in the maximally parallel and in the sequential mode. It turns out that there is a trade-off between maximal parallelism and checking programs: Using checking programs (i.e., priorities on the communication rules in the programs of the agents), P colonies working in the sequential mode with height at most 5 are computationally complete, whereas when working in the maximally parallel mode, P colonies (again with height 5) already obtain the same computational power without using checking programs. Moreover, when allowing an arbitrary number of programs for each agent, we can prove that P colonies with only one agent (thus these P colonies are working in the sequential mode) are already computationally complete. Finally, P colonies with an arbitrary number of agents working in the sequential mode as well as even P colonies with only one agent using an arbitrary number of non-checking programs characterize the family of languages generated by matrix grammars without appearance checking.